Each of the fours steps of the piDMD framework outlined in section  pose distinct challenges.
In some scenarios where the physics is poorly understood, determining suitable physical laws
to impose on the model (step 1, `modeling') can be challenging.
For problems with intricate geometries and multiple dimensions, interpreting the physical principle as a matrix manifold (step 2, `interpretation') can be the roadblock, as the manifold can become exceedingly complicated.
We have already commented on the challenges and opportunities posed by step 3 (`optimization'), and suggested algorithmic solutions [1]}.
Finally, the typically large state dimension and number of samples can obfuscate step 4 when the matrix manifold does not easily admit the desired diagnostics.
For example, if \(A\)  has a complicated banded structure, as in local piDMD, it may not be amenable to a fast eigendecomposition or SVD.
